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Multivariable-Calculus · Laplacian

Laplacian of x^2 - y^2

\nabla^2\left(x^{2} - y^{2}\right)

Step by step

  1. f(x, y) = x^{2} - y^{2},\quad \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}

    The Laplacian adds the second partial derivative in each direction.

  2. \frac{\partial f}{\partial x} = 2 x,\quad \frac{\partial^2 f}{\partial x^2} = 2

    Differentiate twice with respect to x, holding the other variables constant.

  3. \frac{\partial f}{\partial y} = - 2 y,\quad \frac{\partial^2 f}{\partial y^2} = -2

    Differentiate twice with respect to y, holding the other variables constant.

  4. \nabla^2 f = \left(2\right) + \left(-2\right) = 0

    Add the second partials.

  5. \nabla^2 f = 0

    The Laplacian is zero, so f is harmonic: it solves Laplace's equation.

Reveal the answer
\nabla^2 f = 0